A question about Quantum Tunneling

I was wondering about the effect of tunneling of particles in a magnetic field.

Classically, a charged particle will whiz around in a circle in a magnetic field, and barring any interruptions, electric fields or gradients in the magnetic field, it'll stay centered around the same magnetic field line.

Now, I know the gist of tunneling and such, but I'm certainly no expert on the matter. Given that the force is perpendicular to particle motion, I suppose the particle isn't considered to have any potential energy with respect to the magnetic field?

In effect, my question is if due to tunneling effects, an energetic charged particle can slowly move across a magnetic field by virtue of "tunneling" from a given position at a given time to a different position that happens to be extremely close, but still a little farther (or closer) to the field line it was effectively orbiting. My (basic) understanding is that there's always a probability that a particle is actually here, rather than there, so my thought process leads me to wonder if that applies in a magnetic field, which would basically result in transport across field lines.

It's pretty likely that my lack of deep familiarity with governing equations are the reason I can't answer this for myself. I was hoping an expert could give me a quick yes or no answer.